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Concepts, Misconcepts and Difficulties In Teaching and Learning By Carol A. Balfe, Ph.D. INTRODUCTION: In 1993, the American Association for the Advancement of Science (AAAS) published Benchmarks for Science Literacy, an important precursor to the current National Science Education Standards2 (NSES) and the California Science Standards3. Benchmarks recommended important concepts that all students should know upon graduation from high school and broke these recommendations down by grade spans. Benchmarks also provided a review of research of student understanding, common misconceptions and difficulties with science and math concepts. Both the Benchmarks and NSES make similar recommendations with respect to understanding density as a fundamental property of matter. Both documents recommend that, by the end of 8th grade, students should have developed an understanding that all matter is made of atoms and that equal volumes of different substances usually have different weights1. Through teaching examples2 and research citations1, both documents assert that a conceptual grasp of density is essential to students' being able to develop an understanding of the conservation of matter. Benchmarks specifically states that students cannot understand conservation of matter, if they do not understand what matter is, accept weight (mass) as an intrinsic property of matter, or distinguish between weight (mass) and density1. The Oakland4 and California Science Standards3 specifically state what students should know about density conserving the intent of both Benchmarks and NSES. Beyond the call to understand the scientific concepts, both Benchmarks and NSES call for science instruction to interface effectively with mathematics instruction1, 2. Benchmarks in recommending that students explore the notion of function, references the NCTM Standards related to working with patterns.5 Benchmarks recommends that middle school students work with graphs to explore relationships with two variables and to pay close attention to rigorous logic based on observed and carefully documented patterns. The NSES explicitly call for coordination of mathematics and science program "to enhance student understanding of mathematics in the study of science and to improve student understanding of mathematics." NCTM's Standards and Principles likewise exhorts us to "recognize and apply mathematics in contexts outside of mathematics" and to "select, apply and translate among mathematical representations to solve problems." The Oakland and California Science Standards in the Investigation and Experimentation sections explicitly call for students to use the mathematical concepts delineated in this density unit.3,4 The comments and thoughts that follow are intended to help teacher-learners understand some of the struggles of middle school learners as they grapple with notions of matter and density and with the underlying mathematics. HISTORICAL CONTEXT: Historically, the evidence used in developing atomic / molecular theory was complicated and abstract. A recent translation of Sir Isaac Newton's The Principia: Mathematical Principles of Natural Philosophy, originally published in 1686,5 puts the intellectual challenge into perspective. With the following words, Newton opened the monumental treatise that laid the foundation for modern scientific and mathematical thought: "Quantity of matter is a measure of matter that arises from its density and volume jointly." Thus, Newton articulated, for the first time in scientific history, a mathematical relationship between density, volume and "quantity of matter" (which he later called "mass.") In the next few sentences, he clarified his definition with examples and spelled out the underlying implications of his definition: If the density of air is doubled in a space that is also doubled, there is four times as much air, and there is six times as much if the space is tripled. The case is the same for snow and powders condensed by compression or liquefaction, and also for all bodies Newton then adds: Furthermore, I mean this quantity
whenever I use the term "body" or "mass"
in the following pages. It can always be known from a body's
weight, for - by making very accurate measurements with pendulums
- I have found it to be proportional to the weight, as will be
shown below. The commentary to a recent translation
of The Principia6 highlights the profound difficulty of
these concepts. Contemporaries of Newton's described "quantity
of matter" as "extension, or space occupied" (Descartes),
as "moles" or "bulk" (Kepler), or simply
as weight (Galileo). Newton himself struggled with the definition
and arrived at it only in the last stages of writing The
Principia6. The "simple" definition described
above represents years of thoughtful struggle. Yet, today, we
ask middle school students to grapple with, indeed to "master,"
these deceptively complex ideas. Perhaps, we should not be surprised
that young (and not-so-young) learners of the 21st century, finds
these ideas challenging. Underlying the concept of density,
then are some of the following concepts: Newton and Galileo were both mathematicians! Yet, each of their contributions to the development of science rested strongly on their adherence to experimentation as a precursor to mathematical description and interpretation. Middle school students, no less that Galileo or Newton, need to experiment, explore, experience and develop a feel for the physical phenomena. Then, as did Newton and Galileo, they need to learn to observe, record and measure with care and precision. Again, as did Newton and Galileo, they must examine their data, look for the mathematical patterns and relationships. Finally, they must state the relationships, perhaps write them as equations and show how the mathematics elucidates the physical phenomena. This is no mean set of tasks for a 13-year-old! Adult learners, too, find the concepts challenging as reported in Journal of College Science Teaching.7 In describing her college physics classes, the author observes that "students have trouble with the concept of density. Although it is covered in high school chemistry and physics students had little usable knowledge." The author then proceeds to describe efforts to turn this around for her students by giving them hands-on, qualitative and quantitative experiences. So, in the accompanying unit, there are many opportunities to explore qualitatively, the notions of density as a fundamental property of materials and the relationship of volume and mass to one another in a given material. These opportunities include multiple explorations of floating and sinking as well as opportunities to develop a sense of the inverse relationship of mass and volume in "fluffy" materials and "dense" materials. It is important that the students reflect on these qualitative explorations, look for patterns and ask many questions about their observations. To develop a quantitative understanding, students do need to have a working understanding of math concepts of proportion and ratios and to be comfortable with fractions and their decimal equivalents. Middle school math teachers know well, that students find these math ideas challenging and difficult 1,2,5,8. Even those students who develop some facility with ratios, proportions and fractions, usually experience these concepts in the isolated setting of a mathematics classroom. The challenge then, becomes one of helping students to work comfortably with the math concepts and to use them to help make sense of the physical phenomena. Graphing their data can help students to see the mathematical relationship of mass vs. volume more easily. The challenge will be to ensure that the students' measurements are sufficiently precise and accurate that their graphs will aid rather than hinder. Thus, allowing sufficient time for students to measure, learn to measure and correct their errors will lower their frustration. Middle school students commonly have experience with graphing. Many will be more comfortable with making bar graphs than with graphing points on an X-Y plane. This will also represent the first time most of the students have had to examine the relationship between two variables (in this case, mass and volume) to describe a physical property (density.) Students will need practice and exploration time to work out how to scale their graphs (more applications of proportional reasoning). They will need still more time and coaching to learn to interpret their graphs both visually, and ultimately, mathematically. Interpretation should include predicting values not actually measured, as well as realizing that the slope of the graph yields the density. Given an opportunity to explore the meaning of their graphs, students can be helped to realize that the standard "formula" for density (D = mass/volume) arises from graphing mass vs. volume data and determining the slope. If students "play" with their data and their graphs, they can come to realize that larger masses will also have larger volumes, but the mass/volume ratio will always be the same. Obvious as that may seem to one who "gets the concept" of density, this is an "Aha!" moment for students. More importantly, they will repeatedly encounter similar patterns as they continue their studies of the natural world. Exploring density in depth, as outlined
in the accompanying unit, represents a considerable investment
of time and planning on the part of the science teacher. It
requires careful attention to the students' hands-on skills and
no less attention to their mathematical observations and skills.
The time and attention will be well spent. Students, through
this unit, can begin to develop comfort with connecting
measurement, observation and reflection, and the use of mathematics
as a powerful tool to help them make sense of the physical world.
Students will revisit these skills later this year as they explore
motion and forces. They will apply the concept of density as
a fundamental property of matter will be immediately to explore
the patterns of the Periodic Table. 1. Project 2061, American Association for the Advancement of Science, Benchmarks for Science Literacy: Oxford University Press: New York, 1993; pp 77 - 78 and 336 -337. 2. National Academy of Sciences, National Science Education Standards, National Academy Press: Washington, D.C.; 1996; pp 154 and 214. 3. California State Board of Education, Science Content Standards for California Public Schools, California Department of Education, Sacramento, CA, 1998: Grade 8 Standards. 4. Oakland Unified School District, Science Content Standards, Oakland Board of Education, 1997, Eighth Grade Science Standards. 5. National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, The National Council of Teachers of Mathematics, Inc., 2000, pp.210 and ff. and p. 402. 6. Newton, Isaac, The Principia: Mathematical Principles of Natural Philosophy, Translated by I. Bernard Cohen and Anne Whitman, Preceded by "A Guide to Newton's Principia" by I. Bernard Cohen, University of California Press, Berkeley, 1999, pp. 86 -95 and pp. 403 - 404. 7. Roach, Linda, Exploring Concepts of Density, Assessing Nonmajors' Understanding of Physics. J. of College Science Teaching 30(6):386-9. 8. Barnett, Carne, Goldstein, Donna and Jackson, Babette, Mathematics Teaching Cases: Fractions, Decimals and Percents, Hard to Teach and Hard to Learn? Heinemann, Portsmouth, NH. RELATED READING: Children's Ideas in Science, ed. Driver, Rosalind, Guesne, Edith and Tiberghien, Andrée, Open University Press, Bristol, PA, 1985. Osborne, Roger and Freyberg, Peter, Learning in Science: The implications of children's science. Heinemann, Auckland, 1985. Stavy, Ruth and Tirosh, Dina, How Students (Mis-Understand Science and Mathematics, Intuitive Rules, Teachers College Press Columbia University, New York, 2000. NSTA Pathways to the Science Standards: Guidelines for Moving the Vision into Practice, ed. Steven J. Rakow, National Science Teachers Association, Washington, D.C., 1998. |
